AE uses color management, but FCP does not. FCP has no idea how to interpret the color from a movie with a codec like Animation and it defaults to unmanaged color with a gamma of 1.8.

FCP assumes that ProRes 422 and ProRes 4444 media at HD resolutions are Rec. 709, so if you render to ProRes instead of Animation as Joey has suggested, you should have consistent color.

The "Enable Final Cut Studio Compatibility" preference tells QuickTime Player to display color in the same totally inaccurate way that the FCP canvas does (ignoring ColorSync and adjusting the gamma). It doesn't actually affect the media, how FCP interprets it, or how color management-aware applications interpret it.